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About 10,000 papers for “Randomized algorithm”

Randomized Gossip Algorithm with Unreliable Communication

Guodong Shi, Mikael Johansson, Karl Henrik Johansson · arXiv (Cornell University) · 2012

In this paper, we study an asynchronous randomized gossip algorithm under unreliable communication. At each instance, two nodes are selected to meet with a given probability. When nodes meet, two unreliable communication links are established with communication in each direction succeeding with a time-varying probability. It is shown that two particularly interesting cases arise when these communication processes are either perfectly dependent or independent. Necessary and sufficient conditions on the success probability sequence are proposed to ensure almost sure consensus or $ε$-consensus. Weak connectivity is required when the communication is perfectly dependent, while double connectivity is required when the communication is independent. Moreover, it is proven that with odd number of nodes, average preserving turns from almost forever (with probability one for all initial conditions) for perfectly dependent communication, to almost never (with probability zero for almost all initial conditions) for the independent case. This average preserving property does not hold true for general number of nodes. These results indicate the fundamental role the node interactions have in randomized gossip algorithms.

Cited by 4Open access

A Universal Randomized Packet Scheduling Algorithm

Łukasz Jeż · Algorithmica · 2012

We give a memoryless scale-invariant randomized algorithm ReMix for Packet Scheduling that is e/(e−1)-competitive against an adaptive adversary. ReMix unifies most of previously known randomized algorithms, and its general analysis yields improved performance guarantees for several restricted variants, including the s-bounded instances. In particular, ReMix attains the optimum competitive ratio of 4/3 on 2-bounded instances. Our results are applicable to a more general problem, called Item Collection, in which only the relative order between packets’ deadlines is known. ReMix is the optimal memoryless randomized algorithm against adaptive adversary for that problem.

Cited by 13Open access

A fast randomized algorithm for orthogonal projection

Vladimir Rokhlin, Mark Tygert · arXiv (Cornell University) · 2009

We describe an algorithm that, given any full-rank matrix A having fewer rows than columns, can rapidly compute the orthogonal projection of any vector onto the null space of A, as well as the orthogonal projection onto the row space of A, provided that both A and its adjoint can be applied rapidly to arbitrary vectors. As an intermediate step, the algorithm solves the overdetermined linear least-squares regression involving the adjoint of A (and so can be used for this, too). The basis of the algorithm is an obvious but numerically unstable scheme; suitable use of a preconditioner yields numerical stability. We generate the preconditioner rapidly via a randomized procedure that succeeds with extremely high probability. In many circumstances, the method can accelerate interior-point methods for convex optimization, such as linear programming (Ming Gu, personal communication).

Cited by 1Open access

A Randomized Algorithm for Optimal PID Controllers

Yossi Peretz · Algorithms · 2018

A randomized algorithm is suggested for the syntheses of optimal PID controllers for MIMO coupled systems, where the optimality is with respect to the H ∞ -norm, the H 2 -norm and the LQR functional, with possible system-performance specifications defined by regional pole-placement. Other notions of optimality (e.g., mixed H 2 / H ∞ design, controller norm or controller sparsity) can be handled similarly with the suggested algorithm. The suggested method is direct and thus can be applied to continuous-time systems as well as to discrete-time systems with the obvious minor changes. The presented algorithm is a randomized algorithm, which has a proof of convergence (in probability) to a global optimum.

Cited by 18Open access

A Randomized Algorithm for Closest-Point Queries

Kenneth L. Clarkson · SIAM Journal on Computing · 1988

An algorithm for closest-point queries is given. The problem is this: given a set S of n points in d-dimensional space, build a data structure so that given an arbitrary query point p, a closest point in S to p can be found quickly. The measure of distance is the Euclidean norm. This is sometimes called the post-office problem. The new data structure will be termed an RPO tree, from Randomized Post Office. The expected time required to build an RPO tree is $O(n^{\lceil {{d / 2}} \rceil (1 + \epsilon )} )$, for any fixed $\epsilon > 0$, and a query can be answered in $O(\log n)$ worst-case time. An RPO tree requires $O(n^{\lceil {{d / 2}} \rceil (1 + \epsilon )} )$ space in the worst case. The constant factors in these bounds depend on d and $\epsilon $. The bounds are average-case due to the randomization employed by the algorithm, and hold for any set of input points. This result approaches the $\Omega (n^{\lceil {{d / 2}} \rceil } )$ worst-case time required for any algorithm that constructs the Voronoi diagram of the input points, and is a considerable improvement over previous bounds for $d > 3$. The main step of the construction algorithm is the determination of the Voronoi diagram of a random sample of the sites, and the triangulation of that diagram.

Cited by 249

A Randomized Parallel Algorithm for Dfa-Minimization

Ti On · 2007

The problem of finding the coarsest partition of a set S with respect to another partition of S and one or more functions on S has several applications, one of which is the state minimization of finite state automata. The problem has a well known O(n log n) sequential algorithm. In this paper, we present efficient parallel randomised algorithms for the problem. Keywords: Parallel-algorithms, Partitioning, DFA minimization 1 Introduction The single function coarsest partitioning problem can be stated as follows: Problem : Given a set S of n elements, a 1 ; a 2 ; :::; a n , a partition ß = (ß 1 ; ß 2 ; :::; ß k ) of S and a function f : S ! S, we wish to find the coarsest (i.e., having the fewest blocks) partition ß 0 = (ß 0 1 ; ß 0 2 ; :::; ß 0 l ) of S, such that 1. 8a i ; a j 2 S, a i ; a j 2 ß 0 m ) 9p such that, f(a i ); f(a j ) 2 ß 0 p . 2. 8m; 1 m l; 9p; 1 p k; such that ß 0 m ` ß p . The more general version of the problem, is the multiple function coarsest ...

Cited by 0

A Randomized Linear-time Majority Tree Algorithm

Nina Amenta, Frederick Clarke, Katherine St. John · 2008

We give a randomized linear-time algorithm for computing the majority tree, a technique widely used for summarizing sets of phylogenetic trees. We are implementing the algorithm as part of an interactive visualization system for analyzing large sets of trees.

Cited by 0

An Improved Randomized Algorithm for 3-SAT

Rainer Schuler, Uwe Schöning, Osamu Watanabe · 2001

Schöning [Sch99] proposed a simple yet efficient randomized algorithm for solving the k-SAT problem. His analysis showed that for 3-SAT, finding a satisfying assignment of any satisfiable formula F with n variables can be achieved in poly(n) (4/3) n (= poly(n) (1.3333) n ) expected time, which is optimal up to now. In this paper, we improve this expected time bound by using a combination of a deterministic algorithm for 3-SAT and an improvement of the above randomized algorithm. Our new bound for the above task is poly(n) (1.3303) n .

Cited by 0

Randomized Speedup of the Bellman-Ford Algorithm

Michael J. Bannister, David Eppstein · arXiv (Cornell University) · 2011

We describe a variant of the Bellman-Ford algorithm for single-source shortest paths in graphs with negative edges but no negative cycles that randomly permutes the vertices and uses this randomized order to process the vertices within each pass of the algorithm. The modification reduces the worst-case expected number of relaxation steps of the algorithm, compared to the previously-best variant by Yen (1970), by a factor of 2/3 with high probability. We also use our high probability bound to add negative cycle detection to the randomized algorithm.

Cited by 4Open access

A randomized Kaczmarz algorithm with exponential convergence

Thomas Strohmer, Roman Vershynin · ArXiv.org · 2007

The Kaczmarz method for solving linear systems of equations is an iterative algorithm that has found many applications ranging from computer tomography to digital signal processing. Despite the popularity of this method, useful theoretical estimates for its rate of convergence are still scarce. We introduce a randomized version of the Kaczmarz method for consistent, overdetermined linear systems and we prove that it converges with expected exponential rate. Furthermore, this is the first solver whose rate does not depend on the number of equations in the system. The solver does not even need to know the whole system, but only a small random part of it. It thus outperforms all previously known methods on general extremely overdetermined systems. Even for moderately overdetermined systems, numerical simulations as well as theoretical analysis reveal that our algorithm can converge faster than the celebrated conjugate gradient algorithm. Furthermore, our theory and numerical simulations confirm a prediction of Feichtinger et al. in the context of reconstructing bandlimited functions from nonuniform sampling.

Cited by 0Open access

Randomized LOBPCG algorithm with linear dimension reduction

Yanfei Xiang · SPIRE - Sciences Po Institutional REpository · 2025

We present a randomized variant of the locally optimal block preconditioned conjugate gradient (LOBPCG) method that incorporates linear dimension reduction for computing a group of eigenpairs of generalized eigenvalue problems. In contrast to the well-established LOBPCG method, we do not assert that our randomized variant always yields a superior converging eigensolver. Instead, our objective is to offer a preliminary exploration of merging statistical randomization into deterministic LOBPCG, aiming to enhance our understanding of the numerical performance of this randomized LOBPCG variant for solving eigenvalue problems. Additionally, this work paves some potential avenues for integrating dimension reduction techniques into other deterministic numerical linear algebra algorithms. Finally, we empirically demonstrate the performance of this novel randomized LOBPCG variant using extensive academic examples, revealing that its efficiency is comparable to, and in some cases surpasses, original LOBPCG.

Cited by 0Open access

Repeated randomized algorithm for the Multicovering Problem

Abbass Gorgi, Mourad El Ouali, Anand Srivastav et al. · Greater South Information System · 2021

Let $\mathcal{H}=(V,\mathcal{E})$ be a hypergraph with maximum edge size $\ell$ and maximum degree $Δ$. For given numbers $b_v\in \mathbb{N}_{\geq 2}$, $v\in V$, a set multicover in $\mathcal{H}$ is a set of edges $C \subseteq \mathcal{E}$ such that every vertex $v$ in $V$ belongs to at least $b_v$ edges in $C$. Set multicover is the problem of finding a minimum-cardinality set multicover. Peleg, Schechtman and Wool conjectured that unless $\cal{P} =\cal{NP}$, for any fixed $Δ$ and $b:=\min_{v\in V}b_{v}$, no polynomial-time approximation algorithm for the Set multicover problem has an approximation ratio less than $δ:=Δ-b+1$. Hence, it's a challenge to know whether the problem of set multicover is not approximable within a ratio of $βδ$ with a constant $β<1$. This paper proposes a repeated randomized algorithm for the Set multicover problem combined with an initial deterministic threshold step. Boosting success by repeated trials, our algorithm yields an approximation ratio of $ \max\left\{ \frac{15}{16}δ, \left(1- \frac{(b-1)\exp\left(\frac{ 3δ+1}{8}\right)}{72 \ell} \right)δ\right\}$. The crucial fact is not only that our result improves over the approximation ratio presented by Srivastav et al (Algorithmica 2016) for any $δ\geq 13$, but it's more general since we set no restriction on the parameter $\ell$. Furthermore, we prove that it is NP-hard to approximate the Set multicover problem on $Δ$-regular hypergraphs within a factor of $(δ-1-ε)$. Moreover we show that the integrality gap for the Set multicover problem is at least $\frac{\ln_{2}(n+1)}{2b}$, which for constant $b$ is $Ω(\ln n )$.

Cited by 0Open access